Model Photospheres

I. What is a photosphere?

II. Hydrostatic Equilibruium

III.Temperature Distribution in the Photosphere

IV. The Pg-Pe-T relationship

V. Properties of the Models

VI. Models for cool stars

I. What is a Stellar Atmosphere?• Transition between the „inside“ and „outside“ of the star

• Boundary between the stellar interior and the interstellar medium

• All energy generated in the core has to pass through the atmosphere

• Atmosphere does not produce any energy

• Two basic parameters:

• Effective temperature. Not a real temperature but the temperature needed to produce the observed flux via 4R2T4

• Surface Gravity – log g (although g is not a dimensionless number). Log g in stars range from 8 for a white dwarf to 0.1 for a supergiant. The sun has log g = 4.44

What is a Photosphere?• It is the surface you „see“ when you look at a star

• It is where most of the spectral lines are formed

What is a Model Photosphere?

• It is a table of numbers giving the source function and the pressure as a function of optical depth. One might also list the density, electron pressure, magnetic field, velocity field etc.

• The model photosphere or stellar atmosphere is what is used by spectral synthesis codes to generate a synthetic spectrum of a star

Real Stars:

1. Spherical2. Can pulsate 3. Granulation, starspots, velocity fields4. Magnetic fields5. Winds and mass loss

Our model:

1. Plane parallel geometry

2. Hydrostatic equilibrium and no mass loss

3. Granulation, spots, and velocity fields are represented by mean values

4. No magnetic fields

P + dPP

dA

r + dr

M(r)

The gravity in a thin shell should be balanced by the outward gas pressure in the cell

r

dm

A

P +dP

dr

P Gravity

II. Hydrostatic equilibrium

Fp = PdA –(P + dP)dA = –dP dA Pressure Force

FG = –GM(r) dMr2 Gravitational Force

r

0M(r) = ∫(r) 4r2 dr

dM = dA dr

Both forces must balance: FP + FG = 0

–dP dA –G+ (r)M(r)

r2

drdA = 0

dm

A

P +dP

dr

P Gravity

The pressure in this equation is the total pressure supporting the small volume element. In most stars the gas pressure accounts for most of this. There are cases where other sources of pressure can be significant when compared to Pg.

Other sources of pressure:

1. Radiation pressure: PR = 4

3c T4 = 2.52 ×10–15 T4 dyne/cm2

2. Magnetic pressure: PM = B2

8

3. Turbulent pressure: ~ ½v2 v is the root mean square velocity of turbulent elements

Footnote: Magnetic pressure is what is behind the emergence of magnetic „flux ropes“ in the Sun

Pphot

Ptube

In the magnetic flux tube the magnetic field provides partial pressure support. Since the total pressure in the flux tube is the same as in the surrounding gas Ptube < Pphot. Thus tube < phot and the flux tube rises due to buoyancy force.

+ B2

8

Teff

(K)

Sp Pg

(dynes/cm2)

PR

(dynes/cm2)

B

(Gauss)

v

4000 K5 V 1 × 105 0.6 1584 7.5

8000 A6 V 1 × 104 10 501 10.6

12000 B8 V 3 × 103 52 274 13.0

16000 B3 V 3 × 103 165 274 15.0

20000 B0 V 5 × 103 403 354 16.7

Pressures

B is for magnetic pressure = Pg

v is velocity that generates pressure equal to Pg according to ½v2

We are ignoring magnetic fields in generating the photospheric models. But recall that peculiar A-type stars can have huge global magnetic fields of several kilogauss in strength. In these atmospheric models one has to treat the magnetic pressure as well.

dA

F

gravity

dx

P

P + dPF + dF

x increases inward so no negative sign

d = dx

dPd

=g

dP –G (r)M(r)

r2

dr=

In our atmosphere

dP = gdx

The weight in the narrow column is just the density × volume × gravity

GM/r2 = g (acceleration of gravity)

One way to integrate the hydrostatic equation

Pg½ dPg = Pg

½ g0

d00 is a reference wavelength (5000 Å)

Pg(0) = dt0g32 00

∫o Pg

½

(

(⅔

Pg(0) = dlog t0g32 0 log e

–∞∫log o Pg

½

(

(⅔t0½

Integrating on a logarithmic optical depth scale gives better precision

Pg(0) = dt000∫o gPg

½3/223

Numerical Procedure

Problem: we must now as a function of since appears in the integrand. is dependent on temperature and electron pressure. Thus we need to know how T and Pe depend on 0.

• Guess the function Pg(0) and perform the numerical integration

• New value of Pg(0) is used in the next iteration until convergence is obtained

• A good guess takes 2-3 iterations

III. Temperature Distribution in the Solar Photosphere

Two probes of depth:

• Limb Darkening

• Wavelength dependence of the absorption coefficient

Limb darkening is due to the decrease of the continuum source function outwards

I () = ∫Se–sec sec d0

∞

The exponential extinction varies as sec , so the position of the unit optical depth along the line of sight moves upwards, i.e. to smaller .

ds

The increment of path length along the line of sight is ds = dx sec

2

1 dz

=1 surface

Top of photosphere

Bottom of photosphere

Temperature profile of photosphere

100008000

6000

4000

z=0

Tem

pera

ture

z

The path length dz is approximately the same at all viewing angles, but at larger the optical depth of =1 is reached higher in the atmosphere

Limb Darkening

Solar limb darkening as a function of wavelength in Angstroms

Solar limb darkening as a function of position on disk

At 1.3 mm the solar atmosphere exhibits limb brightening

Horne et al. 1981

Temperature profile of photosphere and chromosphere

100008000

6000

4000

z=0

Tem

pera

ture

z

chromosophereIn radio waves one is looking so high up in the atmosphere that one is in the chromosphere where the temperature is increasing with heigth

Limb darkening in other stars

Use transiting planets

At the limb the star has less flux than is expected, thus the planet blocks less light

No limb darkening transit shape

The depth of the light curve gives you the Rplanet/Rstar, but the „radius“ of the star depends on the limb darkening, which depends on the wavelength you are looking at

To get an accurate measurement of the planet radius you need to model the limb darkening appropriately

If you define the radius at which the intensity is 0.9 the full intensity:

At =10000 Å, cos =0.6, =67o, projected disk radius = sin = 0.91

At =4000 Å, cos =0.85, =32o, projected disk radius = sin = 0.52 → disk is only 57% of the „apparent“ size at the longer wavelength

The transit duration depends on the radius of the star but the „radius“ depends on the limb darkening. The duration also depends on the orbital inclination

When using different data sets to look for changes in the transit duration due to changes in the orbital inclination one has to be very careful how you treat the limb darkening.

Possible inclination changes in TrEs-2?

Evidence that transit duration has decreased by 3.2 minutes. This might be caused by inclination changes induced by a third body

But the Kepler Spacecraft does not show this effect.

One possible explanation is that this study had to combine different data sets taken at different wavelength band passes (filters). But the limb darkening depends on wavelength. At shorter wavelengths the star „looks“ smaller.

The only star for which the limb darkening is well known is the Sun

In the grey case had a linear source function:

S = a + b

I () = ∫Se–sec sec d0

∞

Using:

I () = a + b cos

This is the Eddington-Barbier relation which says that at cos = the specific intensity on the surface at position equals the source function at a depth

Ic = Ic(0) (1 – + cos )Limb darkening laws usually of the form:

≈ 0.6 for the solar case, 0 for A-type stars

Ic(0) continuum intensity at disk center

I () = ∫Se–sec sec

0

∞ d log log e

Rewriting on a log scale:

Contribution function

No light comes from the highest and lowest layers, and on average the surface intensity originates higher in the atmosphere for positions close to the limb.

Sample solar contribution functions

Wavelength Variation of the Absorption Coefficient

Since the absorption coefficient depends on the wavelength you look into different depths of the atmosphere. For the Sun:

• See into the deepest layers at 1.6m

• Towards shorter wavelengths increases until at = 2000 Å it reaches a maximum. This corresponds to a depth of formation at the temperature minimum (before the increase in the chromosphere)

Solar Temperature distributions

Best agreements are deeper in the atmosphere where log 0 = –1 to 0.5

Poor aggreement is higher up in the atmosphere

Temperature Distribution in other Stars

The simplest method of obtaining the temperature distribution in other stars is to scale to a standard temperature distribution, for example the solar one.

T(0) = S0Tּס(0)

In the grey case:

T() = [ + q()]¾¼Teff

T() =Teff

Tּסeff

Tּס() In the grey case the scaling factor is the ratio of effective temperatures

Scaled solar models agree well (within a few percent) to calculations using radiative equilibrium. They also agree well when applying to giant stars. Numerically it was easier to use scaled solar models in the past. Now, one just uses a grid of models calculated using radiative equilibrium

When solving the hydrostatic pressure equation we start with an initial guess for Pg(0). We then require that the electron pressure Pe(0) = Pe(Pg) in order to find 0(0) = 0(T,Pe) for the integrand. The electron pressure depends on the temperature and chemical composition.

IV. The Pg–Pe–T Relation

N1j

N0j

=j(T)

Pe

N1j = number of ions per unit volume of the jth element

N0j = number of neutrals

(T) = 0.65 u1

u0

T5/2

10–5040I/kT

See Saha equation from 2nd lecture

Neglect double ionization. N1j = Nej, the number of electrons per unit volume that are contributed by the jth element.

j(T)

Pe

=Nej

N0j

Nej

Nj – Nej=

The total number of jth element particles is Nj = N1j + N0j. Solving for Nej

j(T)/PeNej Nj=

j(T)/Pe

The pressures are:

Pe = NejkTj

Pg = (Nej + Nj)kTj

Taking ratios:

NejkT

(Nej + Nj)kT

Pe

Pg

=

Aj

PePg=

j(T)/Pe

j(T)/Pe

Aj

j(T)/Pe

j(T)/Pe

1 +

NjPe

Pg

=

j(T)/Pe

j(T)/Pe

Nj

j(T)/Pe

j(T)/Pe

1 +

Using the number abundance Aj = Nj/NH

NH = number of hydrogen

This is a transcendental equation in Pe that has to be solved iteratively. (T) are constants for such an iteration. Pe and Pg are functions of 0. This equation is solved at each depth using the first guess of Pg(0).

log

–4

–3–2

0

1

–1

For the cooler models the temperature sensitivity of the electron pressure is very large with d log Pe/d log T ≈ 12 since the absorption coefficient is largely due to the negative hydrogen ion

The absorption coefficient is largely due to the negative hydrogen ion which is proportional to Pe so the opacity increases very rapidly with depth.

Hydrogen dominates at high temperatures and when it is fully ionized Pg ≈ 2Pe

At cooler temperatures Pe ~ Pg½

Where does the later come from? Assume the photosphere is made of single element this simplifies things:

Pe = Pg

(T)/Pe

1 + 2 (T)/Pe

Pe = (T)Pg – 2(T)Pe = (T)(Pg– 2Pe)2

Pg >> Pe in cool stars

Pe ≈(T)Pg

2

Completing the model

Pg(0) = dlog t0g32 0 log e

–∞∫log o Pg

½

(

(⅔t0½

We can now can compute this

• Take T(0) and our guess for Pg(0)

• Compute Pe(0) and 0(0)

• Above equation gives new Pg(0)

• Iterate until you get convergence (≈ 1%)

• Can now calculate geometrical depth and surface flux

We are often interested in the geometric depth scale (i.e. where the continuum is formed). This can be computed from dx = d0/0

The Geometric Depth

x(0) = ∫1

0(t0)(t0)dt0

0

0

The density can be calculated from the pressure ( P = (/)KT )

= NH (hydrogen particles per cm3) xAjj grams/H particle) where j is the atomic weight of the jth element

NH =Aj

N – Ne

kTA

j

Pg – Pe=

x(0) = ∫ AjkT(t0)t0

0(t0) Ajj[Pg(t0)t0 – Pe(t0)]d log t0 d log e

–∞

log 0

x(0) = ∫ AjkT(p)

Ajj

1 g

–∞

Pg

dpp

A more interesting form is to integrate on a Pg scale with dPg = gdx

The thickness of the atmosphere is inversely proportional to the surface gravity since T(Pg) depends weakly on gravity

This makes physical sense if you recall the scale height of the atmosphere:

Scale height H = kT/g

Computation of the Spectrum

The spectrum

F= 2 ∫

∞

StE2t)dt

F= 2 ∫∞

SE2)–∞

(0)0 d log t0

0(0) d log e

It is customary to integrate on a log scale

Flux contribution function

Flux Contribution Functions as a Function of Wavelength

Flux at 8000 Å originates higher up in the atmosphere than flux at 5000 or 3646 Å

But cross the Balmer jump and the flux dramatically increases. This is because there is a sharp decrease in the opacity across the Balmer jump.

Flux Contribution Functions as a Function of Effective Temperature

A hotter star produces more flux, but this originates higher up in the atmosphere

T= 10400 K

T= 8090 K

T= 4620 K

Computation of the Spectrum

There are other techniques for computing the flux → Different integrals.

Integrating flux equation by parts:

F= 2 ∫

∞

SE2)d

F= S(0) + ∫

∞

E3)d d S

dt

The flux arises from the gradient of the source function. Depths where dS/d is larger contribute more to the flux

V. Properties of Models: Pressure

Relationship between pressure and temperature for models of effective temperatures 3500 to 50000 K. The dashed line marks where the slope exceeds 1–1/≈ 0.4 and implies instability to convection

d log T/d log Pg = 0.4

Convection gradient

Tem

pera

ture

Cannot scale T(Pg), unlike T(0)!!!

Teff = 8750 K

Increasing the gravity increases all pressures. For a given T the pressure increases with gravity

dlogPg

dlog g = 0.85

dlogPg

dlog g = 0.62

Effects of gravity

Pg(0) = dlog t0g32 0 log e

–∞∫log o Pg

½

(

(⅔t0½

Pg ≈ C(T) g ⅔ since pressure dependence in the integral is weak. So dlog P/dlog g ~ 0.67

In general Pg ~ gp

In cool models p ranges from 0.64 to 0.54 in going from deep to shallow layers

In hotter models p ranges from 0.85 to 0.53 in going from deep to shallow layers

Recall Pe ≈ Pg½ in cool stars → Pe ≈ constant g⅓

Pe ≈ 0.5 Pg in hot stars → Pe ≈ constant g⅔

Properties of Models: Chemical Composition

• In hot models hydrogen takes over as electron donor and the pressures are indepedent of chemical composition

• In cool models increasing metals → increasing number of electrons → larger continous absorption → shorter geometrical penetration in the line of sight → gas pressure at a given depth decreases with increasing metal content

Gas pressure Electron pressure

Qualitatively:

Using Aj for the sum of the metal abundances

Pe

Pg

A j

=Pe

Pg

N j

kT

kT

=Pe

Pg

P

N jkT≈

Pe

N jkT

Since PH, the partial pressure of Hydrogen dominates the gas pressure

Nj = (N1 + N0)j, the number of element particles is the sum of ions and neutrals and Pe=NekT = N1jkT for single ionizations

A j

Pe

Pg

≈ N 1j

(N 1 + N0)j

In the solar case metals are ionized N1j >> N0j

A j

Pe

Pg

≈

dPg =g0

d0

gPe0/Pe

d0= =g

PgAj0/Pe

0 is dominated by the negative hydrogen ion, so 0/Pe is independent of Pe

Integrating:

Aj

d0

g

0/Pe½Pg

2= ∫

0

0

g and T are constants

Aj)Pg =c0

–½ Aj)Pe =c0

½

For metals being neutral: N1J << (N1 + N0)j can show

Aj)Pg =c0

–⅓ Aj)Pe =c0

⅓

Properties of Models: Effective Temperature

Note scale change of ordinate

• In hotter models opacity increases dramatically

• More opacity → less geometrical penetration to reach the same optical depth

• We see less deep into the stars → pressure is less

• But electron pressure increases because of more ionization

This is seen in the models

If you can see down to an optical depth of ≈ 1, the higher the effective temperature the smaller the pressure

Properties of Models: Effective Temperature

For cool stars on can write:

Pe ≈ C eT

At high temperatures the hydrogen (ionized) has taken over as the electron donor and the curves level off

A grid of solar models

Log

Tem

pera

ture

Depth (km)

Dep

th (

km)

Log

The mapping between optical depth and a real depth

Wavelength (Ang)

Am

plit

ude

(mm

ag)

Why do you need to know the geometric depth?

In the case the pulsating roAp stars, you want to know where the high amplitude originates

Log

Log

(Pre

ssur

e)

Pgas

Pelectron

Pe ≈Pg½

Note bend in Main Sequence at the low temperature end. This is where the star becomes fully convective

VI. Models for Cool Objects

Models for very cool objects (M dwarfs and brown dwarfs) are more complicated for a variety of reasons, all related to the low effective temperature:

1. Opacities at low temperatures (molecules, incomplete line lists etc.) not well known

2. Convection much stronger (fully convective)

3. Condensation starts to occur (energy of condensation, opacity changes)

4. Formation of dust

5. Chemical reactions (in hotter stars the only „reactions“ are ionization which is give by ionization equilibrium)

Much progress in getting more complete line lists for water as well as molecules. Models have gotten better over time, but all models produce a lack of flux (over opacity) in the K-band.

M8V

Allard et al. 20101971

2001

1995

1997

Dust Clouds

The cloud composition according to equilibrium chemistry changes from:

Zirconium oxide (ZrO2) Perovskite and corundum (CaTiO3, Al2O3)

Silicates: forsterite (Mg2SiO4) Salts: (CsCl, RbCl, NaCl)

Ices: (H2O, NH3, NH4SH)

M → L → T dwarfs

At Teff < 2200 K the cloud layers become optically thick enough to initiate cloud convection.

Teff = 2600 K : No dust formation

Teff = 2200 K : dust has maximum optical thick density

Teff = 1500 K: Dust starts to settle and gravity waves causing regions of condenstation

Allard et al. 2010

Intensity variation due cloud formation and granulation.

Of course these models do not include rotation and Brown Dwarfs can have high rotation rates. Jupiter is as a good approximation as to what a brown dwarf atmosphere really looks like

Most cool star models have a use a more complete line lists for molecules, including water, and also include dust in the atmosphere

Teff = 2900 K, log g=5-0 model compared to GJ 866

8000 Å

Discrepancies are due to missing opacities10000 Å4000 Å

3m

Optical

Infrared

Comparison of Models

NextGen: overestimating Teff

Ames-Cond/Dusty: underestimating Teff

BT-Settl: Using Asplund Solar abundances

Stars with „normal“ opacities

Condensation

Dust clouds

Allard et al. 2010

Now days researchers just download models from webpages. Kurucz model atmospheres have become the „industry standard“, and are continually being improved. These are used mostly for stars down to M dwarfs. The Phoenix code is probably more reliable for cool objects.

• Kurucz (1979) models - ApJ Supp. 40, 1

R.L. Kurucz homepage: http://kurucz.harvard.edu

• The PHOENIX homepage

P.H. Hauschildt:

http://www.hs.uni-hamburg.de/EN/For/ThA/phoenix/index.html

• Holweger & Müller 1974, Solar Physics, 39, 19 – Standard Model

• Allens Astrophysical Quantities (Latest Edition by Cox)

TEFF 5500. GRAVITY 0.50000 LTEITLE SDSC GRID [+0.0] VTURB 0.0 KM/S L/H 1.25OPACITY IFOP 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0CONVECTION ON 1.25 TURBULENCE OFF 0.00 0.00 0.00 0.00BUNDANCE SCALE 1.00000 ABUNDANCE CHANGE 1 0.91100 2 0.08900ABUNDANCE CHANGE 3 -10.88 4 -10.89 5 -9.44 6 -3.48 7 -3.99 8 -3.11ABUNDANCE CHANGE 9 -7.48 10 -3.95 11 -5.71 12 -4.46 13 -5.57 14 -4.49ABUNDANCE CHANGE 15 -6.59 16 -4.83 17 -6.54 18 -5.48 19 -6.82 20 -5.68ABUNDANCE CHANGE 21 -8.94 22 -7.05 23 -8.04 24 -6.37 25 -6.65 26 -4.37ABUNDANCE CHANGE 27 -7.12 28 -5.79 29 -7.83 30 -7.44 31 -9.16 32 -8.63ABUNDANCE CHANGE 33 -9.67 34 -8.69 35 -9.41 36 -8.81 37 -9.44 38 -9.14ABUNDANCE CHANGE 39 -9.80 40 -9.54 41 -10.62 42 -10.12 43 -20.00 44 -10.20ABUNDANCE CHANGE 45 -10.92 46 -10.35 47 -11.10 48 -10.18 49 -10.58 50 -10.04ABUNDANCE CHANGE 51 -11.04 52 -9.80 53 -10.53 54 -9.81 55 -10.92 56 -9.91ABUNDANCE CHANGE 57 -10.82 58 -10.49 59 -11.33 60 -10.54 61 -20.00 62 -11.04ABUNDANCE CHANGE 63 -11.53 64 -10.92 65 -11.94 66 -10.94 67 -11.78 68 -11.11ABUNDANCE CHANGE 69 -12.04 70 -10.96 71 -11.28 72 -11.16 73 -11.91 74 -10.93ABUNDANCE CHANGE 75 -11.77 76 -10.59 77 -10.69 78 -10.24 79 -11.03 80 -10.95ABUNDANCE CHANGE 81 -11.14 82 -10.19 83 -11.33 84 -20.00 85 -20.00 86 -20.00ABUNDANCE CHANGE 87 -20.00 88 -20.00 89 -20.00 90 -11.92 91 -20.00 92 -12.51ABUNDANCE CHANGE 93 -20.00 94 -20.00 95 -20.00 96 -20.00 97 -20.00 98 -20.00ABUNDANCE CHANGE 99 -20.00EAD DECK6 72 RHOX,T,P,XNE,ABROSS,ACCRAD,VTURB2.18893846E-03 2954.2 6.900E-03 1.814E+06 6.092E-05 1.015E-02 0.000E+00 0.000E+00 0.000E+002.91231236E-03 2989.5 9.180E-03 2.404E+06 6.205E-05 9.083E-03 0.000E+00 0.000E+00 0.000E+003.83968499E-03 3098.1 1.211E-02 3.187E+06 6.586E-05 5.843E-03 0.000E+00 0.000E+00 0.000E+005.04036827E-03 3099.5 1.590E-02 4.145E+06 6.588E-05 6.526E-03 0.000E+00 0.000E+00 0.000E+006.60114478E-03 3220.4 2.082E-02 5.367E+06 6.923E-05 4.064E-03 0.000E+00 0.000E+00 0.000E+008.62394743E-03 3221.5 2.722E-02 6.977E+06 6.963E-05 4.606E-03 0.000E+00 0.000E+00 0.000E+001.12697597E-02 3324.7 3.558E-02 8.957E+06 7.213E-05 3.199E-03 0.000E+00 0.000E+00 0.000E+001.47117179E-02 3325.9 4.649E-02 1.165E+07 7.271E-05 3.535E-03 0.000E+00 0.000E+00 0.000E+001.92498763E-02 3393.3 6.080E-02 1.503E+07 7.447E-05 2.979E-03 0.000E+00 0.000E+00 0.000E+002.51643328E-02 3425.0 7.950E-02 1.949E+07 7.579E-05 2.870E-03 0.000E+00 0.000E+00 0.000E+003.29103373E-02 3462.1 1.040E-01 2.526E+07 7.737E-05 2.733E-03 0.000E+00 0.000E+00 0.000E+004.30172811E-02 3502.2 1.359E-01 3.272E+07 7.906E-05 2.600E-03 0.000E+00 0.000E+00 0.000E+005.62014022E-02 3540.2 1.776E-01 4.238E+07 8.090E-05 2.473E-03 0.000E+00 0.000E+00 0.000E+007.33634752E-02 3576.1 2.318E-01 5.487E+07 8.294E-05 2.359E-03 0.000E+00 0.000E+00 0.000E+009.56649214E-02 3609.9 3.023E-01 7.100E+07 8.521E-05 2.267E-03 0.000E+00 0.000E+00 0.000E+001.24571186E-01 3643.2 3.936E-01 9.176E+07 8.778E-05 2.219E-03 0.000E+00 0.000E+00 0.000E+001.61931251E-01 3676.7 5.117E-01 1.184E+08 9.071E-05 2.191E-03 0.000E+00 0.000E+00 0.000E+002.10049977E-01 3710.3 6.637E-01 1.525E+08 9.409E-05 2.159E-03 0.000E+00 0.000E+00 0.000E+002.71775682E-01 3744.2 8.588E-01 1.959E+08 9.802E-05 2.123E-03 0.000E+00 0.000E+00 0.000E+003.50602645E-01 3779.0 1.108E+00 2.511E+08 1.026E-04 2.087E-03 0.000E+00 0.000E+00 0.000E+004.50760286E-01 3814.6 1.424E+00 3.209E+08 1.080E-04 2.071E-03 0.000E+00 0.000E+00 0.000E+005.77317822E-01 3851.4 1.824E+00 4.086E+08 1.142E-04 2.062E-03 0.000E+00 0.000E+00 0.000E+007.36336604E-01 3889.2 2.327E+00 5.186E+08 1.216E-04 2.038E-03 0.000E+00 0.000E+00 0.000

Sample grid from Kurucz